The Hammers could move into the top four with a win over Crystal Palace on Tuesday night as they look to make it six victories in a row across all competitions.
West Ham have more points (32) to their name at the halfway point of the 2020/21 season than they have had at this stage in any other Premier League campaign.
And Moyes doesn't want to stop now.
With this season like no other meaning the race at the top is as tight as ever, there are hopes that the east Londoners can break into the top six come the end of the season.
Asked if that was his ambition, Moyes said: "I might tell the players that behind closed doors, but I certainly won't come out and publicise it because I think that that is how you get shot down. But I am trying to be ambitious, I want us to move forward and I think we have done that already this season.
"We have made big strides forward already, and then sometimes folk want even more. I want more but I know we can only take small steps and we are in amongst a lot of incredibly good sides, there are quite a lot of good sides below us as well and we are going to have to play really well to keep them away from us."